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Theoretical Seismology 2: Wave Propagation
Thailand Training Program in Seismology and Tsunami Warnings, May 2006
Theoretical Seismology 2: Wave Propagation
Seismic waves in an elastic medium
・ Rays and Ray Paths
How they propagate in the Earth
Travel-time curves
・ Near-Field Terms (Static Displacements)
• Far-Field Terms (P, S, Surface waves)
• Surface Waves
・ Normal modes
(Free oscillations of the Earth)
Explosion
P waves
Psychology test
Homogeneous Earth
(seismologist)
(engineer)
(tsunamicist?)
Ray Paths in a Layered Medium
Snell’s law: sin a1 / sin a2 = a1 / a2
a1 q1 Slower a1 q1 Faster
q2
Faster q2 Slower
a２ a２
a1 < a2 a1 > a2
Travel-time Curves for Travel-time curves
Ray Paths in a Layered Medium
Time
1/a3
1/a2
1/a1
Distance
(D=T/velocity)
a1 head wave
a2
a3
Velocity structure of the Earth
Ray Paths in a Gradient Travel-time curves
Time
Velocity gradient can be treated as a
series of thin homogeneous layers. 1/a3
1/a2
1/a1
Distance
(D=T/velocity)
a1
a2
a3
Moho
Andrija Mohorovicic (1857-1936)
Found seismic discontinuity at
30 km depth in the Kupa Valley
(Croatia).
Mohorovicic discontinuity or ‘Moho’
Boundary between crust and mantle
Structure in the Earth
Conrad and Moho Discontinuities
Low velocity zone
Forward Branch
Receding Branch
Forward Branch
Shadow Zone
Forward Branch (PKPbc)
Receding Branch (PKPab)
PcP
Receding
Branch A
Forward PKP
Branch B C
Forward
Branch
Shadow
PcP Shadow
Zone P Zone
Forward Branch
Receding Branch Forward
Branch
Not shown: PKP(DEF) and PKiKP
Other notation for core phases:
ABC branch known as P2l
DEF branch known as P1ll
PKP(DEF) known as PKIKP
Point B is a caustic
PcP
Core Reflections
Faulting
Seismic waves
Other aspects of wave propagation:
• Diffracted Waves
・ Surface Waves
・ Static Displacements
・ Frequency content
• Normal Modes
Other aspects of wave propagation
• Diffracted Waves
・ Surface Waves
・ Static Displacements
・ Frequency content and wavelength
• Normal Modes
1-D Wave Equation
u1 1 u1
2 2
x12
c t 2
1-D wave equation
c = propagation speed
2 u1 1 2 u1
x1 2
c t 2
Solution
u( x, t ) A sin[ (t x / c)]
2 T = wave period
T
= angular frequency
LW 3.2.1
Wave Period and Wavelength
Velocity = Wavelength / Period
Space x
Velocity 6 km/s
wavelength
period 50 s
Wavelength 300 km
Time
t
period 50 s
frequency = 1/period= 0.02 hz
period
Period Wavelength
Body waves 0.1 to 50 sec 50 m to 500 km
（P・S）
Surface waves 10 to 350 sec 30 to 1000 km
Free Oscillations 350 to 3600 sec 1000 to 10000 km
Static
Displacements
-
Other aspects of wave propagation
•Diffracted waves
・ Surface waves
・ Static Displacements
(amplitude at zero frequency)
・ Frequency content
• Normal modes
3-D Wave Equation with Source
2u
2 f ( 2 )( u ) ( u )
t
source spatial 2nd derivative
Near-field Terms (Static Displacements)
Solution
1 1 r/ 1 1 r 1 1 r
u ( x, t ) a M 0 (t )d A IP M 0 (t ) A IS 2 M 0 (t )
N
A 4
4 r r/ 4a 2 r2 a 4 2 r
1 1 r 1 1 r
A FP M 0 (t ) A FS M 0 (t )
4a 3 r a 4 3 r
Far-field Terms (P, S Waves)
Near-field terms
・ Static displacements
r/a r/
・ Only significant close to the fault
・ Source of tsunamis
r/a r/
t →
Static Displacements
Bei-Fung Bridge near Fung-Yan city, 1999 Chi-Chi, Taiwan earthquake
Static displacements
Co-seismic deformation
of 2003 Tokachi-oki
Earthquake (M8.0)
Generation of Tsunami from Near-field Term
EA
PAC
Far-field Terms
1
A FP
1 r
M 0 (t )
1 1 r
A FS M 0 (t )
4a 3
r a 4 3
r
・ Propagating Waves
・ No net displacement
in an elastic medium
・ P waves
・ S waves
Other aspects of wave propagation
• Diffracted Waves
・ Surface Waves
・ Static Displacements
・ Frequency content
• Normal Modes
Surface Waves
Group Velocity (km/sec)
Love
Rayleigh
S Period (sec)
Shearer, Fig. 8.1
January 26, 2001 Gujarat, India Earthquake (Mw7.7)
Body waves
vertical
Rayleigh Waves
P PP S SS
radial
transverse
Love Waves
Recorded in Japan at a distance of 57o (6300 km)
Other aspects of wave propagation
• Diffracted Waves
・ Surface Waves
・ Static Displacements
・ Frequency content
• Normal Modes
Free Oscillations of the Earth
(Normal Modes)
Few minutes after the earthquake Few hours after the earthquake (0S20)
Constructive interferences free oscillations
(or stationary waves)
Standing Waves with Periods < 54 min, amplitudes < 1 mm
Observable months after great earthquakes (e.g. Sumatra, Dec 2004)
From Michel van Camp, Royal Obs. of Belgium
Normal Modes
(Stein and Gellar 1978)
Free Oscillations of the Earth (Daishinji, Fukui Prefecture)
1960 Chile Earthquake
Useful for studies of
・ Interior of the Earth
・ Largest earthquakes
Toroidal and Spheroidal Modes
Toroidal
Spheroidal
Dahlen and Tromp Fig. 8.5, 8.17
Natural Vibrations of the Earth
Indexes describe spherical harmonics
Shearer Ch.8.6
Lay and Wallace, Ch. 4.6
Free Oscillations l=1 m=1
Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html
Free Oscillations l=1 m=2
Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html
Free Oscillations l=1 m=3
Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html
Structure: Free Surface
Earth is a not homogenous whole-space
Free surface causes many complications
- surface waves
- reflections (pP, sP, sS)
Summary
Rays
Velocity structure includes gradients, discontinuities
and LVZ’s, causing complicated ray paths
through the Earth (P, PKP, PcP)
Wave theory explains
・ P and S waves
・ Static displacements
・ Surface waves
Normal Modes
The Earth rings like a bell at long periods
Why are observed seismograms so
messy ?
